Small-Signal Modeling of Pulse-Width
Modulated Switched-Mode Power
Converters
R. D. MIDDLEBROOK,
FELLOW, IEEE
Invited Paper
A power processing system is required to convert electrical
energy from one voltage, current, or frequency to another with,
ideally, 100-percent efficiency, together with adjustability of the
conversion ratio. The allowable elements are switches, capacitors,
and magnetic devices. Essential to the design is a knowledge of
the transfer functions, and since the switching converter is nonlinear, a suitable modeling approach is needed. This paper discusses
the state-space averaging method, which is an almost ideal compromise between accuracy and simplicity. The treatment here
emphasizes the motivation and objectives of modeling by equivalent circuits based upon physical interpretation of state-space averaging, and is limited to pulse-width modulated dc-to-dc converters.
I.
INTRODUCTION
Almost all electronic equipment contains a power supply, often with a regulated output. The so-called linear regulator, originally predominant, has largely given way to the
switching regulator having higher efficiency and smaller
sizeand weight. Despite these obvious benefits, the switching power supply unfortunately also has some disadvantages, including conducted and radiated noise arising from
the switching process.
Less apparent is the fact that a switching power supply
presents an order of magnitude greater difficulty in design.
Both linear and switching power supplies are feedback systems, and therefore can be characterized in terms of the
familiar concepts of loop gain, bandwidth, and stability
margins. Analysis, and hence design, of a linear regulator
can be accomplished with use of standard linear circuit
techniques incorporating small-signal linearized models of
the nonlinear active devices. In contrast, the switching
power supply feedback loop contains a modulator and a
switching power stage that constitute a type of analog-todigital followed by a digital-to-analog conversion function,
Manuscript received September21,1987; revised January6,1988.
This work was conducted in part under the Power Electronics Program supported by GTE Communication Systems Corporation,
EG&C Almond Instruments, and Rockwell Inc.
The author i s with the Power Electronics Group, California Institute of Technology, Pasadena, C A 91125, USA.
I E E E Log Number 8820080.
for which, at the outset, there was no model for analysis of
the transfer functions.
Over more than 15 years, much has been achieved in
modeling the nonlinear switching converter power stages
and modulators, and also in application of the models to
the design of improved, more cost-effective switching regulators.
Rather than attempt a review of the many approaches
which, in the available space, would inevitably result in a
superficial treatment of all of them, this paper instead will
emphasize the motivations and objectives of modeling by
one approach, state-space averaging, that represents an
almost ideal tradeoff between accuracy and simplicity,
which no doubt accounts for i t s widespread adoption.
In order to make the topic more easily accessible to
nonspecialists, thescopewill beeven more limited: the bulk
of this paper emphasizes the motivation and objectives of
modeling by equivalent circuits, since these give physical
insight into the qualitative behavior and are an essential
prerequisite to proper use of the more abstract analytic and
computational methods. Indeed, historically, the equivalent circuit approach preceded the generalization embodied in state-space averaging.
For the specialist reader, the paper offers a perspective
on the equivalent circuit models, and their comparative
properties for different operating modes, together with a
summary of the formal state-space averaging method.
The work of the author, his colleagues, and students of
the Power Electronics Group at the California Institute of
Technology has been collected in [I]. Much of it concerns
the subject of modeling, and illustrates the development
of the viewpoint summarized here. Many other workers,
some of whom are explicitly referenced, have also contributed to the subject of modeling, particularly in extensions of the state-space averaging method, as well as different approaches.
In Section II, a power processing system i s compared and
contrasted with a signal processingsystem. Not onlyare the
objectives different, but so are the allowable elements from
which they may be built.
0018-9219/88/0400-0343$01.00 0 1988 IEEE
P R O C E E D I N G S OF THE IEEE, VOL. 76, NO. 4, APRIL 1988
343
In Section Ill, a generalized dc-to-dc regulated power
supply i s used as a vehicle for introduction of the motivation and objectivesof mode1ing.A switched-mode power
stage and a "linear" power stage are both nonlinear systems, and the modeling objective is the same for each: to
find a small-signal ac model whose element values are constant but functions of the large-signal dc operating point.
In Section IV, canonical models are established in a qualitative manner for pulse-width-modulated switched-mode
converters in various modes of operation. A canonical
model is merely an equivalent circuit that has a fixed configuration, regardless of the type of converter it represents,
although its element values are different for different converters.
A qualitative comparison of the properties of the various
operating modes for different converters is made in Section
V. This is the purpose of modeling, so that the designer i s
awareof the natureofthetransferfunctions to bedealtwith
in design of the regulator feedback loop.
Thedetailed foundation of the modeling described in the
earlier sections i s state-space averaging, reviewed in Section VI. Equivalent circuit models are one of several useful
forms of the results; others are analytic expressions for frequency responses, and numerical outputs of individual
transfer functions.
Conclusions are drawn in Section VII. The state-space
averaging method does not accommodate another important class of power processing systems, namely resonant
converters, and methods for their analysis are not mentioned in this paper; nor are modeling methods for dc-toac inverters.
II. DESIGNCONSTRAINTS
AND OBJECTIVES
OF POWER
PROCESSINGSYSTEMS
Power processing systems, contrasted with signal processing systems, are required to process power with minimum power loss, rather than to process information with
minimum power consumption. The function of electrical
power processing systems is to convert electrical energy
from one voltage, current, or frequency to another. This
function is to be achieved, ideally, with 100-percent efficiency. In most applications, including those of greatest
technical interest, the conversion is to be accomplished
adjustably; this issothataregulatedoutputcan beobtained
by closing afeedback loop that causes the conversion ratio
to be automatically adjusted.
The term power processing suggests an intentional contrast with signal processing, a function much more familiar
to most modern electronics engineers. As indicated in Fig.
l(a), a signal processing system is concerned with performing operations upon the information content of an input:
typical functions are analog amplification, digital recoding,
analog-to-digital and digital-to-analog conversions. To perform these functions, electrical power i s required, usually
in dc form and in an amount to be minimized.
In a power processing system, as indicated in Fig. l(b),
the "power" and "information" inputs are interchanged,
and the concern i s with performing operations upon the
power input according to the functions specified by the
information input. Typical functions are dc-to-dc conversion, dc-to-ac inversion, ac-to-dc rectification, and ac-to-ac
cycloinversion.
The distinctions between signal and power processing
344
I
PFiOCESSED
rn
-nn-nrL
(a)
PROCESSED
PROCESSING
SYSTEM
N FO
?
J%%
t
A
ti++
b
V
(b)
Fig. 1 . A signal processing system (a), contrasted with a
power processing system (b).
systems extend down to the most basic level of design: the
selection of elements available for use by the engineer. In
somewhat oversimplified summary, these are resistors,
inductors, capacitors, and active devices that can be used
either in linear (analog) modeor in switching(digita1) mode.
All these elements are available t o the signal processing
integrated-circuit designer-except the inductors. Magnetic elements are shunned because they are bulky, heavy,
expensive, and do not (yet) fit conveniently on to semiconductor chips. Indeed, it i s convenient t o develop, when
necessary, quite complicated circuitryto simulate inductor
properties. Of course, what i s being simulated i s Ldildt, the
signal processing function, not i L i 2 , the energy storage
function needed for power processing.
In contrast, the power processing designer has at his disposal a different array of elements: resistors are now anathema because they are lossy and do not meet the efficiency
requirement. For the same reason, linear operation of active
devices must be excluded, which leaves only inductors,
capacitors, and switches as allowable elements. Of course,
transformers and a wide variety of more complex magnetically coupled structures are also included, and, indeed,
exploitation of these possibilities i s one of the greatest current challenges in the power electronics field [2].
It i s seen that the different set of available elements
requires the power processing designer to develop a quite
different way of thinking from that of the signal processing
designer. One aspect of this i s the recognition of a conflict
unique to the power processing field: minimization of the
parasitic resistances of the magnetics, capacitors, and
switches, desirable in the interests of efficiency, leads also
to undesirable minimum damping of the inherent LCfilters,
with consequent increased difficulties in design of thefeedback loop and in the dynamic range. As usual, proper definition of the problem constitutesa large part of its solution:
what i s needed is lossless damping, of which more will be
said later.
One means of converting one voltagekurrent level to
another is the conventional transformer. Such a practical
device can approach very closely the ideal 100-percent efficiency, especially in large sizes. However, from the point
of view of power processing, it suffersfrom two severe limitations: it does not change frequency, and it does not operate at dc. Although its capabilities are limited, the physical
PROCEEDINGS OF THE
IEEE,
VOL. 76, NO. 4, APRIL 1988
transformer. The most general objective of the power processing design may then be crystallized thus: How t o find
an optimum arrangement of the allowed circuit elements,
as suggested in Fig. 2(b), i n order t o achieve the conversion
functions shown in Fig. 2(a)?Part of the answer lies in knowledge of the small-signal ac properties of the various possible configurations.
transformer nevertheless suggests a characterization of the
general power conversion functions, namely, conversion
of one voltage/current/frequency combination t o another,
in which dc is included as zero-frequency ac.
I n turn, this generalization suggests a useful representation, or model, of the power conversion function: the circuit symbol shown i n Fig. 2(a), an adjustable-ratio ideal
I l l . MOTIVATION
AND OBJECTIVES
OF MODELING
POWER PROCESSING SYSTEM
OBJECTIVE:
Of the four processing functions designated in Fig. l(b),
the modeling approach for only one, dc-to-dc conversion,
will be discussed in this paper. The essence of such a dc
regulated power supply i s shown i n Fig. 3. For illustration,
the switchingconverter isa simple buck (voltage step-down)
circuit; any other converter, such as the basic boost (voltage
step-up) or flyback (step-up or step-down), shown in Fig. 4,
could be substituted. In all of these, the transistor and diode
operate as a single-pole switch.
In Fig. 3, the unregulated line input at voltage vg passes
through a switching converter t o the output of voltage v,
from which an output current i i s drawn requiring a line
inputcurrenti,.Thetransistor switch iscontrolled byaduty
ratio(digita1) signal d f r o m theoutputof a modulator,which
is usually a simple comparator whose inputs are a clock
sawtooth or triangular waveform and an (analog) control
signal. In the waveforms shown here, the modulator turns
o n the transistor switch uniformly at the beginning of each
clock cycle, and turns it off when the control signal overtakes the ramp. If the control signal varies, so does the duty
ratio. This constant frequency, pulse-width modulated
(pwm) transistor switch, together with the power diode,
present a square voltage waveform to the LC filter, which
in turn delivers the average component of this waveform
to the output with small residual switching frequency ripple.
control
Adlustable-ratio ideal transformer:
de-to-de, d e - t o - a c , oc-lo-de, oc-to-ac
---I(a)
REALIZATION:
4
conirol
Swltched- mode conversion: approaches
100% efficiency
(b)
Fig. 2. (a) Objective of a power processing system; (b) its
realization by an optimal arrangement of the allowable elements.
V
onolog
MODULATORPOWER-STAGE
SUBSYSTEM
"""gy
L
clock ramp
/
refereweb
output
spectrum
t
bondposs
/
f
2f
3f
f,-f
fs
zfs
3fs
frequency
Fig. 3. A dc-to-dc switching regulator incorporating a three-port duty ratio programmed
modulator-power-stage subsystem whose transfer functions aredefined in terms of ratios
of small-signal ac quantities (hats) superimposed upon large-signal dc quantities (capitals).
The spectrum of the output signal contains the switching frequency, the control frequency, their respective harmonics, and sidebands.
MIDDLEBROOK: MODELING OF POWER CONVERTERS
345
Pf
A
Fig. 4. Left: basic boost converter. Right: basic flyback
(buck-boost)converter.
The regulator feedback loop i s closed via the sensed converter output which i s compared with the reference, and
the (analog) amplifier error signal constitutes the control
signal t o the modulator. The error amplifier output i s represented by a Thevenin equivalent voltage source v, and
series impedance (diamonds are used to distinguish dependent from independent sources represented by circles).
The ultimate analysis objective i s t o find, as functions of
frequency, the loop gain and hence the closed-loop properties of the regulator. The essential prerequisite is t o find
the transfer functions of the three-port subsystem inside
the large box in Fig. 3, designated the modulator-powerstage subsystem. The most important of these are the control-to-output and line-to-output transfer functions and,
secondarily, the converter output and input impedances.
If the subsystem inside the boxes in Fig. 3 were linear, the
control input, represented by the sum of dc and ac components v, = V,
V,, would lead t o a corresponding converter output v = V Pin which the ac component V would
be at the same frequency as 9, and proportional to it, so that
the control-to-output ac transfer function would be W,,
independent of theamplitudeof V,.Thesamewould betrue
of any transfer function.
However, the subsystem inside the box i s not linear;
nevertheless, the objective i s still to find the transfer functions, albeit subject to some approximation. There will be
a price t o pay for the resulting simplification, namely,
restrictions on the validity of the result.
It is to be noted that this problem is not peculiar to the
switching converter subsystem. If the subsystem i s a socalled linear regulator, exactly the same problem exists [3].
A transistor operated in the active region i s a nonlinear
device, and the solution t o the problem lies in finding a
model that represents the small-signal ac properties at a
given large-signal operating point. The resulting model, for
each transistor, contains elements whose values may be
functions of operating point, but are otherwise independent of the ac signal amplitude as long as it is small enough.
There are many such models, known as the tee, hybrid-*,
y-parameter, etc. In the tee model, for example, the value
of the emitter resistance is inversely proportional t o the
emitter dc operating current, but i s taken to be constant
with respect to sufficiently small ac signals.
It i s simple, yet significant, to say that the sole purpose
of an equivalent circuit model i s t o give the right answer
for the transfer functions. In the case of the switching converter subsystem in Fig. 3, the required transfer functions
are those of the three-port subsystem, and so an equivalent
circuit model representing the whole box, not just one transistor, i s required. The objective, however, i s the same: to
find a linear equivalent circuit that represents the properties of terminal small-signal ac variations, in which the
nonlinearities are relegated t o the variation of the element
+
346
+
values with large-signal dc operating point. The operating
point i s defined by the dc values Vg, I, V,, etc.
In addition to the nonlinearity that causes harmonics of
the control signal frequency f t o occur in the output, a
switching converter presents an additional problem: the
presence in the output of the switching frequency fs and
its harmonics, and also sidebands fs - f, etc. The spectrum
of theoutput voltage i s also illustrated in Fig. 3. Fortunately,
this problem i s not as severe as it may seem, because the
basic requirement for low output switching ripple requires
that the low-pass filter corner frequency be well below the
switching frequency fs by at least a decade. This then highly
accentuates the component of the output at the control
fundamental frequency f, and de-emphasizes all the other
frequencies.
Thus, the linearized part of the transfer properties consists of the output signal (magnitude and phase) at the same
frequency as the control signal P,. Hence, as before, a control-to-output transfer function for the linearized system
can be defined as O/P,, and similarly for the other transfer
functions.
Suitable models from which the transfer functions can
be found are discussed in the next section.
Iv.
SMALL-SIGNAL CANONICAL
MODELS
FOR DC-TO-DC
CONVERTERS
With the motivation and objectives of the model of the
modulator-power-stage subsystem in Fig. 3 now clearly
delineated, establishment of the model itself is but a final
short step, at least qualitatively.
As already seen from the description of the operation, the
control-to-output transfer function vh, can be considered
made u p of two parts, the modulator transfer function dlv,
and the power stage transfer function vld. The first part is
very simple, being merely a proportionality factor W,,,,
where V, is the height of the clock ramp. This follows
because the duty ratio d i s zero when the control voltage
v, i s zero, i s unity when the control voltage is V,, and i s
linear in between if the ramp i s linear. The ramp need not
belinear, inwhichcase V,istheequivalentheightofararnp
having the same slope as the actual ramp at a given operating point.
The model of the power stage can be established by introduction of components that represent the three essential
features of any dc-to-dc pwm switching converter, all of
which have already been mentioned.
The first essential feature is the dc-to-dc conversion property itself, already represented by the generalized ideal
transformer symbol of Fig. 2(a), characterized by the conversion ratio M.
The second essential feature is the presence of a low-pass,
essentially lossless filter, since in anydc-to-dc converter the
switching frequency i s t o be heavily attenuated in the output. This feature may be represented directly by a low-pass
LC filter.
The third essential feature i s adjustability of the conversion ratio, which i s implicit in the dependence of M upon
the duty ratio d. This i s inconvenient with respect to application of the model, because when d i s varied according
to the control voltage v, then M = M(d) = M(D d ) also
varies and makes the transformer a nonlinear element. In
accordance with the procedure for linearizing a nonlinear
+
PROCEEDINGS OF THE IEEE, VOL. 76, NO. 4, APRIL 1988
element described in Section Ill, the transformer i s characterized by a conversion ratio fixed at the value M(D) corresponding to agiven operating-point duty ratio D, and the
d variation component i s separated out into two "modulation'' generators proportional t o d, where the proportionality factors are functions of large-signal operating
point.This process, introduced in [4] and described in detail
in [5] and [6], involves the usual small-signal restriction in
order to eliminate the nonlinearity. Both a current and a
voltage modulation generator are necessary, because a
variation of the conversion ratio affects both current and
voltage.
The three model components representing the three
essential features are assembled in Fig. 5 [4]. A further
Fig. 5. Canonical model for pwm converter in duty ratio
programmed continuous conduction mode. This model
contains the three essential features: dc-to-dc conversion
transformer, low-pass filter, and modulation generators.
reduction i s also included: the simple proportionalityof the
modulator transfer function has been inserted into the
modulation generator coefficients, which are now driven
by 0, instead of by d. The total control voltage v, = V,
0, i s thus separated into two parts, and the dc component
V, determines the dc (operating point) duty ratio D = V,/
V, of which the transformer conversion ratio is a function.
This function M(D) is different in different converters, and
may be nonlinear. It may be noted that the transformer symbol is modified by imposition of a straight and a wavy line,
as a reminder that it represents a generalized conversion
function that extends t o zero frequency.
Qualitatively, the model i n Fig. 5 represents the transfer
function properties of any dc-to-dc pwm switching converter, regardless of its actual configuration (topology), and
is therefore designated a canonicalmodel. The values of the
elements, and their dependences upon operating point do,
however, depend o n the converter type and, more drastically, o n the converter mode of operation.
+
A. Canonical Model for Duty-Ratio Programmed
Continuous Conduction Mode
In the description of Fig. 3, it was mentioned that the transistor and diode operate as a single-pole switch. Further
examination reveals, however, that the single-pole switch
can operate in either a two-position or a three-position
mode, per cycle of the switching frequency.
When the transistor is ON, the filter inductor is connected t o the line input, the diode i s OFF, and the inductor
current ramps u p (essentially linearly if the filter corner frequency i s much lower than the switching frequency, which
i s always the case in pwrn converters). When the transistor
MIDDLEBROOK: MODELING OF POWER CONVERTERS
i s turned OFF by the modulator, the inductor current commutates to the diode, which therefore at the same time i s
automatically turned ON. The inductor current thereupon
ramps down. If the inductor current has not reached zero
at theend of theswitching period when the modulator again
turns the transistor ON, a continuous conduction mode
(ccm) of the inductor current exists i n which the transistor
and diode are alternately ON and OFF and operate as a twoposition single-pole switch. The converter then constitutes
a two-switched network.
In ccm, the canonical model of Fig. 5 [;1is useful directly
because the transformer conversion ratio is afunction only
of duty ratio D. Thus, the control voltage V, uniquely determines the duty ratio D, which can be described as dutyratio
programming, and in turn the duty ratio D uniquely determines the converter output voltage V(for a given line voltage Vg) so that, overall, the converter can be described as
voltage programmed i n that the control voltage uniquely
determines the output voltage.
Quantitatively, i n ccm, the specific dependences of the
canonical model element values upon converter type are
as follows. I n the simple buck converter chosen for illustration in Fig. 3, the filter in the model directly represents
the physical filter; however, in someconverters, notablythe
boost and the buck-boost (also known as the flyback) the
inductance in the model has an effective value related t o
the physical inductor value, but i s also a function of operating point.
In the buck converter, both modulation generators, while
dependent upon operating point, are independent of frequency; however, again notably i n the boost and flyback
converters, the voltage generator is also a function of frequency in the form of a right half-plane (rhp) zero, which
i s also a function of operating point.
B. Canonical Model for Duty-Ratio Programmed
Discontinuous Conduction Mode
As described above, the continuous conduction mode
exists if, when ramping down during the transistor OFFtime, the inductor current does not reach zero. Conversely,
i f the inductor current does reach zero before the modulator turns the transistor O N again, the diode current tries
t o reverse but cannot, and the diode thereupon turns OFF.
Following the transistor ON, diode OFF and transistor OFF,
diodeON intervals, there i s therefore athird interval during
which both transistor and diode are OFF and the inductor
current remains at zero. The transistor and diode therefore
operate as a three-position single-pole switch per cycle of
the switching frequency, and the converter then constitutes a three-switched network described as the discontinuous conduction mode (dcm) of the inductor current. Transition from continuoustodiscontinuous modeoccurswhen
theconverter load current falls below some minimumvalue,
usually designed to be about 10 percent of the maximum
load current.
In dcm, thecanonical model of Fig. 5 is not useful directly
becausetheelement values becomestrongfunctionsof the
load current. In particular, the transformer conversion ratio
becomes a function not only of the duty ratio D, but also
of a parameter K = 2fsL/R, so that M = M(D, K ) , where L i s
the filter inductance value and R is an operating point
parameter defined as the ratio of the dc load voltage to the
dc load current. For a resistive load, R i s the same as the load
347
resistance. Because of the dependence of M upon R, it i s
n o longer convenient t o model the conversion ratio by a
transformer since the current and voltage transfer properties become unequal. Forthisand other reasons it is more
useful to represent the discontinuous conduction mode by
a y-parameter model, as shown in Fig. 6.
"standard,"a different modulator connection, which i s not
duty ratio programmed, has rapidly gained acceptance over
the last ten years [9].
Variously known as "current," "current-mode," or "current-reference" programmed, this configuration is shown
in Fig.7.The principal differenceisthatthefixedclock ramp
j
'h
"
fhpflop
Fig. 6 . Canonical model for pwm converter in duty ratio
programmed discontinuous conduction mode, and also in
current-programmed continuous conduction mode. The
output current is only approximately programmed by the
control voltage, since the resistive component of the output
admittance is comparable to the load resistance.
The model of Fig. 6 [8] is again acanonical model, because
it represents qualitatively the properties of any converter
regardless of i t s detailed topology. The two dependent generators y21and yI2 are unequal, so cannot be replaced by
a transformer, confirming that the network is nonreciprocal. The two modulation generators remain, albeit in a different format as y2cand ylc. The inductance, however, is
notably absent explicitly; this i s because the inductor current, in discontinuous conduction, starts and ends each
switching cycle at zero, and therefore drops out as a state
variable of the system. It is, however, implicitly present via
the dependence of some of the element values upon the
parameter K. So i s the switching frequency implicitly
present through the same parameter K, whereas it was
absent altogether from the canonical model for continuous
conduction mode, other than as an upper frequency limit
upon the validity of the model as a whole.
Both ccm and dcm modes of operation areduty ratio programmed, in that the duty ratio i s uniquely determined by
the modulator control voltage. For the ccm, the canonical
model of Fig. 5 shows that the transformer conversion ratio
is solely a function of duty ratio and so, as already mentioned, the output voltage is also programmed by the control, because the output resistance i s contributed only by
parasitic losses.
O n the other hand, in dcm, the resistive component of
the output admittance y22in the canonical model of Fig. 6
is not small, but in fact i s comparable to the resistive component of the load. Consequently, in this mode the converter cannot be said to be either voltage or current programmed.
C. Canonical Model for Current-Programmed Continuous
Conduction Mode
Any duty ratio programmed pwm switching converter
topology having a modulator as shown in Fig. 3 may operate
in either the continuous or discontinuous inductor current
mode. Although such configurations are still considered
348
I
I
I
l/
--stabilizing
I
ramp
current-programmed modulator-power-stage
subsystem incorporatinga minor current feedback loop, and
a stabilizing ramp derived from the clock.
Fig. 7. A
input to the modulator (comparator) i s replaced by a signal
proportional to the ramp waveform of the power stage
switch current, which i s also the current through an inductor regardless of the power stage topology. The clock, as
before, initiates the switch ON-time uniformly at the
switching frequency fs.However, the switch ON-time is now
terminated when the ramping-up inductor current reaches
avalue proportional to the control signal v,. Clearly, a physical "minor" current feedback loop exists within the modulator-power-stage subsystem, inside the "major" voltage
feedback loop of the regulator of Fig. 3.
Instability of the minor loop occurs for a duty ratio 0.5
or greater, unless a stabilizing ramp of suitable slope is
applied t o the modulator input [9], [IO]. This stabilizing ramp
i s easily derived from the clock, as indicated in Fig. 7.
The immediate consequence of the minor feedback loop
i s that its sensed "output," the switchhnductor current,
tends to follow its "input," the control signal v,; in other
words, this current is programmed. As a first approximation, therefore, the entire subsystem in the box of Fig. 3
could be modeled (at least as far as the control-to-output
transfer function i s concerned) by a current generator
across the output, dependent upon the control signal.
It turns out, however, that this model i s rather too simple.
One approach to a more detailed treatment [IO] i s to start
with the canonical model for the duty ratio programmed
power stage, and to add to it the model of the modulator
obtained from analysis of the comparator input waveforms
of Fig. 7. An example i s shown in Fig. 8, for the boost converter in continuous conduction mode.
PROCEEDINGS OF THE IEEE, VOL. 76, NO. 4, APRIL 1988
1
i
C
-
Fig. 8. Basic model of the current-programmed boost converter, showing the current loop via f,,, closed around the
canonical model for the duty ratio programmed mode.
The modulator transfer function &Oc i s n o longera simple
proportionality, since the modulator now has two additional inputs, the inductor current and the linevoltage.The
line voltage appears because in the boost converter, during
the switch ON-time, the line voltage appears across the
inductor and determines the slope of i t s current ramp. I n
the buck converter, during the switch ON-time, the difference between the line and output voltages appears
across the inductor, and consequently the output voltage
i s yet another input t o the modulator.
The model of Fig. 8, or the corresponding one for another
converter, can be analyzed for the loop gain of the minor
current feedback loop. I n all cases it i s found t o be quite
low [IO], i n the range 1 to 10 at low frequencies and, moreover, declines below unity at a crossover frequency fc which
is between one-sixth and two-thirds of the switching frequency.
Although the model of Fig. 8 could be used directly in the
analysis and design of the complete regulator, it i s simpler
first t o reduce this model t o a canonical form [IO], [Ill. In
thisway, thecurrent feeback i s absorbed into thecanonical
model, which in turn can be used as a component in the
analysis of the regulator major voltage feedback loop in the
usual manner. Also, such a canonical model of the subsystem box of Fig. 3 exposes the properties of the power
stage independently of the major voltage loop and, in particular, shows how successfully the current programming
causes the switchlinductor current t o followthe control signal-not very successfully, it turns out.
As already mentioned, a first-order model of the currentprogrammed power stage consists merely of an output current generator dependent upon the control signal. It i s reasonable, therefore, to choose a more general model that
includes such a generator as one of its elements. The
obvious choice i s the y-parameter model.
Hence, the y-parameter model of Fig. 6 i s suitable not only
MIDDLEBROOK: MODELING OF POWER CONVERTERS
t o represent the duty ratio programmed subsystem of Fig.
3 i n discontinuous conduction mode, but also the currentprogrammed subsystem i n continuous conduction mode.
The performance similarity between the two modes even
extends quantitatively: the resistive component of y22 in
current-programmed mode is also o n the order of the output operating point parameter (the load resistance in the
case of resistive load), as it i s for dcm, which means that the
output i s not a stiff current source,which in turn means that
the "current programming" i s not particularly effective.
There are two reasons why the current feedback loop,
now buried inside the y-parameter model, does not lead t o
astiff current sourceatthe power stageoutput.One reason,
already mentioned, is that the current loop gain is quite
low, so that the sensed current does not track the control
signal veryclosely.Theother reason i s thatthecurrent being
sensed, the switch/inductor current, is not necessarily the
power stage output current that is being modeled by the
y-parameters. I n the buck converter, the inductor current
istheoutputcurrent, but in the boost and flyback,forexample, it i s not (because the inductor i s not in series with the
output), and so there i s an additional transfer function
between the inductor current and the output current that
also leads t o a less stiff current source in the model of the
output properties.
It may be concluded that the power stage and modulator
subsystem of Fig. 7 hardly deserves the name "current-programmed," since the current programming is n o more
effective than in the case of the duty ratio programmed discontinuous conduction mode. I n other words, both subsystems are about equally well (or equally poorly) current
programmed, at least in comparison with the stiff voltage
programming that occurs in the duty ratio programmed
continuous conduction mode of operation. It is, however,
too late to suggest any change in the established descriptions.
The canonical models can easily be extended t o represent the properties of many derived converter topologies,
such as the push-pull, forward, and other transformer-coupled versions [12].
V. COMPARISON
OF PROPERTIES
OF DIFFERENT
MODES
OF
OPERATION
The purpose of the canonical models established i n the
previous section is t o permit easy analysis of the various
transfer functions, and to provide insight into the qualitative behavior. This is an intermediate step towards determination of the regulator major loop gain and, hence, the
closed-loop properties of the regulator that must meet
design specifications.
I n this section, a qualitative comparison i s made of the
properties represented bythe threecanonical models, with
respect toonlythe most important of thetransfer functions,
the control-to-output.
The three models discussed are for duty ratio programmed continuous and discontinuous inductor current
modes of operation, and for current-programmed continuous mode of operation. It has already been seen that the
second and third modes havequite similar properties, both
represented by the y-parameter model of Fig. 6. There i s a
fourth mode, current-programmed discontinuous inductor current, but its properties are similar to those of the second and third modes, and will not be discussed.
349
Thecontrol-to-output transfer function O/O,for duty ratio
programmed ccm can be found from the canonical model
of Fig. 5. For zero line source impedance, the current modulation generator is shorted out, and the voltage modulation generator thus appears across the conversion transformer primary. The function O/Q,therefore consistsof three
factors: thecoefficient of thevoltage modulation generator,
the conversion ratio M, and the filter voltage transfer function.
In the case of a buck converter power stage, the voltage
modulation generator coefficient i s independent of frequency and hence the frequency response of the controlto-output transfer function i s simply that of the filter. Also
for the buck, the filter effective inductance L, i s equal to the
physical inductor value L, and so the filter characteristic is
constant, independent of operating point.
Typically, the LCfilter has a Q of at least 0.5, and is therefore underdamped and has a transfer function characterized by a complex pole pair. The resultant peaking i n the
response i s undesirable for several reasons: the accompanying steep phase lag in the control-to-output transfer
function causes stability difficulties i n the regulator major
voltage loop gain, and also the peaking restricts the available dynamic range of ac load current components.
These properties are the core of the design conflict
already alluded t o in Section I I : the lower the losses, the
higher the Q, and the less desirable the dynamic response.
Hence, higher efficiency involves greater dynamic design
difficulty[l3]. One resolution ofthisconflictcan beachieved
by placing adc-blocked damping resistanceacross the filter
capacitor; this i s an example of lossless damping [14].
Both the boost and flyback power stages, in continuous
conduction mode, exhibit even greater design difficulties.
The voltage modulation generator in the canonical model
of Fig. Scontainsan rhpzero, sothecontrol-to-outputtransfer function contains the rhp zero in addition to the two
(complex) poles of the filter. The 270° asymptotic phase lag
severely restricts the allowable bandwidth of the regulator
voltage loop gain, since the rhp zero can lie considerably
below the switching frequency. Worse, not only does the
rhp zero change with operating point, but so do the filter
poles, because the effective inductance itself becomes a
function of the duty ratio.
In duty ratio programmed dcm, the y-parameter canonical model of Fig. 6 shows thatthecontrol-to-output transfer
function i s considerably simpler, for all basic power stage
topologies. The only model elements needed to solve for
O M c are y2c and y22; although both depend on operating
point, neither isafunctionoffrequency,and s o ~ ~ ~resisi s a
tance R,, whose value, as already mentioned, i s comparable
with the converter output operating point parameter R E
VI/. Hence, for a resistive load RL, OM, i s a single-pole function in which the pole i s determined by R2211RL in parallel
with the output capacitance C.
Although the single-pole response is much easier to handle in the regulator loop design than is the high-Q doublepole response, there are disadvantages of the discontinuous compared t o the continuous conduction mode. Since
the inductor current starts and ends each switching cycle
at zero, its rms value i s always higher in relation to its dc
value than in continuous conduction, with consequent
higher losses. Also, there i s higher relative noise. For these
reasons, use of discontinuous conduction mode i s usually
limited to converters for relatively low power ratings.
350
Converters operating in current-programmed continuous mode are also represented by the y-parameter canonical model of Fig. 6, although the expressions for the element values are somewhat more complicated. In particular,
all six elements contain a pole at the current feedback loop
crossover frequency f,,which i s a manifestation of the usual
result that the effects of feedback disappear above the loop
gain crossover frequency.
The pole at fc in the output admittance y2>canbe ignored,
because at this frequency the output capacitance Calready
dominates the total load on the y2, and yZcgenerators. The
remaining resistive component R2* of yZ2,as already mentioned, i s comparable to the operating point parameter R
= VI/ and hence, for a resistive load RL, as in the case of the
discontinuous conduction mode, there i s a pole in the control-to-output transfer function $/$, determined by R221/R,
in parallel with the output capacitance C.
In current-programmed mode, however, there i s a second pole in O/OCthat comes directly from the pole fc in the
y2,generator of Fig. 6. Hence, the control-to-output transfer
function in current-programmed mode is a two-pole function, although one pole i s dominant (that due to C), and the
other pole i s at a much higher frequency (fc, between onesixth and two-thirds of the switching frequency) [IO]. In
other words, the two poles are characterized by a low Qfactor, much less than 0.5 and therefore heavily damped.
Current programming, consequently, introduces the desirable effect of lossless damping, lossless because the damping results from the minor current feedback loop, and not
from physical (lossy) resistance. This so-called ”single-pole
response” i s commonly quoted as one of the principal benefits of current programming; however, “dominant-pole
response” would be a more appropriate term, since the
additional phase lag from the second pole can have a significant influence on the phase margin of the regulator
major voltage feedback loop.
A comparison of the most important features of the
canonical models for the three operation modes that have
been discussed i s displayed in Table 1. The “approximately
current programmed” description refers to the fact, already
discussed, that the output resistance i s comparable to the
value of the (resistive) load, and i s therefore relatively high
compared to that of astiff voltage source, although not high
enough to constitute a stiff current source. The onlyfeature
not so far mentioned i s that the rhp zero in the boost or
flyback converter i s present in the current programmed as
well as duty ratio programmed continuous conduction
modes. This is because the duty ratio programmed model
i s embedded in the current programmed model, as seen in
the exampleof the boost converter in Fig. 8, and thecurrent
minor feedback loop does not affect any zeros in the control-to-output transfer function.
It i s seen from Table 1 that the dynamic properties of the
current programmed mode of operation are in between
those of the duty ratio programmed continuous and discontinuous modes. Moreover, the current programmed
modecombinestheadvantagesoftheother two modes, the
continuous conduction operation together with the dominant pole response, for the price only of retention of the
rhp zero in the boost and flyback converters.
There are two other significant advantages of the current
programmed mode of operation, not directly related to the
small-signal response: implementation of the current sense
inherently provides almost all the components necessary
PROCEEDINGS OF THE IEEE, VOL. 76, NO. 4, APRIL 1988
Table 1 Relative Properties of the Control-to-OutputTransfer Function of pwm
Converters in Three Modes of Operation
Control-to-Output
Transfer Function
rhp zero in
Boost and
Flyback
Operation Mode
output
Poles
Duty ratio programmed
continuous inductor
current mode
Duty ratio programmed
discontinuous inductor
current mode
Current programmed
continuous inductor
current mode
voltage programmed
complex pair
(high Q)
Ye5
approx. current
programmed
single
no
approx. current
programmed
real pair, one
Yes
for a cycle-by-cycle current limit o n the power switch, and
the high output resistance permits paralleling of power
stages inside the same voltage regulator loop, greatly
enhancing the designability of modular power supplies. All
these features have led t o current programming becoming
the preferred mode of operation for most pwm switching
converters.
VI. THESTATE-SPACEAVERAGING
METHOD
The description of converter small-signal models given
in the previous section is a circuit-oriented approach, aimed
at equivalent-circuit results having elements that retain
direct physical interpretation. Indeed, this i s the way in
which the modeling was originally done. However, a formal, general analysis method has been developed [15], [7l
by which the dc and small-signal transfer functions can be
obtained directly without the use of an equivalent circuit
at all, although such a circuit model can be extracted from
an intermediate step.
This general analysis method is state-space averaging. I t s
essence lies in the fact that any pwm converter is a special
kind of nonlinear system, one which i s switched sequentially among two or more linear circuits according t o one
or more duty ratios. Furthermore, the "inputs" or control
signals include not only independent voltages and currents, but also the duty ratios.
The basic buck, boost, flyback, and derived converters
(such as the forward, push-pull, half- and full-bridge) are
two-switched network converters when operated in continuous inductor current mode, in that the system i s
switched back and forth between two linear systems under
the control of the duty ratio. The same converters operated
in discontinuous inductor current mode [8], are threeswitched network systems, since the switching can be represented by a single-pole three-position switch. Only one
of the three intervals i s under independent control; the ratio
between the remaining two intervals (defined by the inductor current falling t o zero) i s not an independent variable.
However, there are other, more elaborate pwm converters
that are multiple-switched networks in which all the duty
ratios are independent control inputs.
Although state-space averaging accommodates all these
cases, only the simplest, that applicable t o two-switched
networks in ccm, will be summarized here. The name itself
is almost self-explanatory.
In each of the two positions of the switch, the system is
linear and state-space equations can bewritten i n the usual
MIDDLEBROOK: MODELING OF POWER CONVERTERS
dominant
(low Q)
way. The state variables are the inductance currents and
capacitance voltages, and a state-space equation i s a firstorder differential equation. I n matrix notation, the two sets
of state-space equations are
+ BTU
x = A2x + B2u
X =
A1x
(1)
(2)
where x is the vector of state variables, U i s the vector of
independent sources (the linevoltage, for example), andAl,
B,, and A2, B2 are respective system matrices in each of the
two switched networks.
The key concept in state-space averaging is the replacement of the two sets of state-space equations by a single
equivalent set
X =
AX
+ Bu
(3)
in which the equivalent matricesA and Bare weighted averages of the actual matrices that alternately describe the
switched system. For a duty ratio d, the system spends a
fraction d of the switching period described by Al and B,,
and the remaining fraction (1 - d) described by A2 and BZ.
Hence [15], [7l, the equivalent matrices are defined by
A
E
dAl
+ (1 - d)A2
(4)
B
E
dB1
+ (1 - d)B2.
(5)
A single equivalent network can be found that i s characterized by the matrices A and B.
With the switched network now described by the equivalent matrices A and B, the solution of (3) now proceeds i n
the usual manner. The steady-state solution, with dc values
indicated by capital letters, i s obtained by setting x = 0:
X = -A-lBU.
(6)
The ac small-signal solution i s found by the substitution U
= U 0, x = X
9 into (31, i n the usual way. However, the
second key step peculiar t o the switching converter application now arises, which involves recognition that the duty
ratio d i s also modulated and constitutes an input; hence,
the substitution d = D
is also made i n (3). Since this
makes (3) nonlinear, the small-signal restriction becomes
necessary in order that the equation may be linearized by
omission of terms in the products of ac quantities. When
this i s done, and the dc terms are subtracted out, the resulting state-space average equation for the small-signal ac
quantities i s
+
+
+2
2 = A)?
+ BO + BdG
(7)
351
where
Bd(A1 - AJX
+ (B,
-
B>)U.
(8)
By Laplace transformation, the solution of (7) can be written
f(s) = ( s l
-
A)-’ [BG(s)
+ B,&s)]
(9)
where I is the identity matrix.
It often happens that the output signal required for some
transfer function is not one of the state variables, but some
combination of them and, in general, some direct function
of the inputs, including the duty ratio. For example, if aconverter output capacitor equivalent series resistance i s
accounted for, the converter output voltage is a function
of both thecapacitance voltage and inductancecurrent state
variables. In such cases, an output vector y may be
expressed for each of the two switched networks as
+ DIU
(10)
y = C2x + D,u
(11)
y = C,X
and weighted averaging of the matrices is done in the same
way as for A and B. The same steps then lead to the results
Y=CX+DU
P(s)
=
C,t(s)
+ D@s)+ Dd&)
(12 )
(13)
where the matrix coefficients are defined by (4), (5),and ( 8 )
with C, D, and Ddsubstituted for A, B, and Bd, respectively.
Equations (12) and (13) provide, in closed form, the dc
conditions and the small-signal frequency response of any
pwm switching converter. The extension to more than two
switched network systems is straightforward.
One of the uses of the results i s that a single equivalent
network can be found that is characterized by the averaged
equations, which i s the generalization of the circuit-oriented approach to modeling that was taken in the earlier
sections. In particular, the averaging step corresponds to
replacement of the switch by the conversion transformer,
and the linearization step requiring the small-signal restriction corresponds to replacement of the conversion transformer varying ratio by a fixed-ratio transformer and the
modulation generators. Note that the matrix coefficients B,
and D,,of dare functions of operating point, as are the coefficients of the modulation generators in the circuit model.
The equivalent circuit models can be used either for analytic prediction of transfer functions, or can be presented
as inputs to computer circuit analysis programs. If the program is capable of handling nonlinearities, the modulation
of the conversion transformer can be accommodated
directly without separation into the modulation generators
1161.
Identification of an equivalent circuit model is, however,
not necessary; the analytic results can be used directly to
solve for any transfer function either numerically [I71 or
algebraically.
The state-space average modeling method is an almost
ideal compromise between accuracy and simplicity. Both
the equivalent circuit format, and the analytic results
embodied in (12) and (13), are in formsfamiliar to electronic
circuit engineers, and are therefore easily accessible. The
principal limitation i s that the results break down for modulation frequencies approaching the switching frequency,
and indeed half the switching frequency i s also a special
case not fully accounted for in the results. However, as dis-
352
cussed earlier, this is exactlythefrequency rangeoverwhich
the results are needed, and extension of the range of validity requires rapidly increasing complexity of the model.
Specifically, as discussed in [7], the inherent constraint
required for the averaging step to be valid i s that the corner
frequency of the converter filter must be much lower than
the switching frequency, by at least an order of magnitude.
This condition i s normally met in pwm converters, but resonant converters belong to a different class for which this
condition i s not met. State-spaceaveraging is therefore not
applicable, at least not directly, to resonant converters, and
their models are not discussed in this paper.
VII.
CONCLUSIONS
Signal processing systems achieve desired functionswith
use of active devices, resistors, and capacitors, but inductors are avoided because of their incompatibility with integrated circuit fabrication techniques. Power processing
systems, on the other hand, must avoid resistors in the
interests of efficiency, and can incorporate only switches,
capacitors, and magnetic structures.
A power processing system, such as a switched-mode dcto-dc regulated power supply, i s also a signal processing
system in that the regulator feedback loop must be characterized by appropriate transfer functions, which include
those of the switched-mode modulator and power stage.
Since the power stage i s nonlinear, one seeks a suitable
model from which the transfer functions can be obtained.
Asdiscussed in Section I l l , a suitable model is asmall-signal
ac model whose element values depend upon the large-signal dc operating point, but are otherwise constant.
Canonical models, of a given configuration but which
have different element values for different converters and
operating conditions, are developed qualitatively in Section IV. Fig. 5 shows the model for duty ratio programmed,
pu Ise-width-modulateddc-to-dcconverters operated i n the
continuous inductor current mode, in which the switching
devices function as a two-position, single-pole switch. The
model contains the three essential features of any such converter: the basic dc-to-dc conversion property, represented
by the conversion transformer symbol which defines
response down to dc; the effective low-pass filter property;
and the adjustability property. Actually, adjustability is
attained by modulation of the duty ratio D, of which the
conversion ratio M i s a function; however, the model is linearized by representation of the adjustability property by
separate voltage and current modulation generators. A duty
ratio programmed converter i s one whose duty ratio drive
is a unique function of the control voltage, as delivered by
a modulatorthatconsistsof acomparator driven bythecontrol voltage and a fixed clock ramp, illustrated in Fig. 3.
Fig. 6 shows the canonical model for the same duty ratio
programmed modulator as in Fig. 3, but with a power stage
operated in the discontinuous inductor current mode, in
which the switching devices function as a three-position
single-pole switch. The same canonical model of Fig. 6 also
applies to a current-programmed power stage, i n which a
local, or minor, current feedback loop is closed around the
modulator-power-stage subsystem of Fig. 3, as illustrated
in Fig. 7. The same y-parameter canonical model of Fig. 6
is appropriate for both modes of operation, because in both
the output current follows the control voltage 9, and i s
therefore modeled by a voltage-controlled current gener-
PROCEEDINGS OF THE I E t E , VOL. 76, NO. 4, APRIL 1988
ator ~ ~ However,
~ 0 again
~ in
. both modes of operation, the
output current follows the control voltage only poorly, and
the resistive component of the output admittance yZ2is not
negligibly large, but comparable to the load resistance.
Strictly speaking, neither mode of operation deserves the
title “current-programmed,” but this i s the name conventionally adopted to describe the configuration of Fig. 7 in
which an explicit attempt i s made to achieve that result.
Comparison of the transfer functions as predicted by the
canonical models for the three modes of operation is
described in Section V, and summarized in Table 1.The current-programmed mode has dynamic-response properties
in between those of the other two, having the major advantages of both, namely dominant pole response in the continuous inductor current mode, but with the disadvantage
of a right half-plane zero in the boost and flyback type converters. Dominant-pole response (low Q, overdamped) is
far preferable to the complex-pole (high Q, underdamped)
response of the duty ratio programmed continuous inductor current mode since the damping i s lossless, which considerably eases the efficiency versus dynamic response
design tradeoff. This advantage, together with the simplicitywith which switch current limit and parallelingof power
stages can be achieved, makes current-programming the
preferred mode of operation.
State-space averaging i s the formal process of which the
canonical equivalent circuit models are one form of the
results. The method is summarized in Section VI. There are
two key steps. One i s the averaging step itself, in which the
state-space equations describing the two or more switched
networks are averaged according to the fractions of the
switching period that the system resides in each configuration. The validity depends upon the corner frequency of
the power stage filter being much lower than the switching
frequency. The second key step accommodates the switch
duty ratios as control inputs, subject to the small-signal limitation. Other useful formsofthe resultsareanalyticexpressions for frequency responses, and numerical outputs of
individual transfer functions [18].
Many authors have discussed the state-space averaging
method [19]-[21], its limitations, and have made extensions
to higher frequencies [22], larger signals [23], and to dc-toac converters [24]. Its major constraint, however, is that it
does not apply to resonant converters, which constitute an
important class for practical applications.
State-space averaging i s not the only analysis method for
pwm switching converters. Some alternative approaches
are discussed in [25]-[30]. Simulation and computer-aided
methods have also been employed [31]-[33].
[I41
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H. A. Owen, A. Capel, and J. G. Ferrante, in Proc. IEEE Power
Electronics Specialists Conf., pp. 45-55, 1976.
F. C. Y. Lee, R. P. Iwens, Y. Yu, and J. E. Triner, "Generalized
computer-aided discrete-time modeling and analysis of dc-dc
converters," IEEE Trans. lndust. Electron. Contr. Instrum., vol.
IECI-26, pp. 58-69, May 1979.
R. D.Middlebrook (Fellow, IEEE) received the
Ph.0. degree in electrical engineeringfrom
Stanford University, Stanford, CA, in 1955.
He i s Professor of Electrical Engineering
at the California Institute of Technology,
Pasadena, CA. His publications include
numerous papers, a book on solid-state
device theory, and another on differential
amplifiers. His research interests, formerly
in solid-state device modeling, are now in
circuits and systems, and particularly in
power processing electronics, in which he is well known as author,
lecturer, and consultant. He is especially interested in design-oriented circuit analysis and measurement techniques which he
teaches at Caltech, and has conducted short courses on his methods in both Europe and the United States.Among his outside interests is Formula race car driving.
Dr. Middlebrook i s the recipient of the 1982 IEEE William E. Newell Power Electronics Award for OutstandingAchievement in Power
Electronics, and a 1982 Award for Excellence in Teaching, presented by the Board of Directors of the Associated Students of Caltech.
PROCEEDINGS OF THE IEEE, VOL. 76, NO. 4, APRIL 1988